Tomographic Reconstruction Using Information-Weighted Spline Smoothing

1 I n t r o d u c t i o n The CBP method for tomographic reconstruction is derived from a mathemat i cal idealization of tomographic imaging without consideration of statistical measurement errors. This idealization leads to the well-known ramp filter [1], whose frequency response has the unfortunate effect of amplifying high-frequency measurement noise. The conventional approach to reducing the effects of measurement noise is to apodize the ramp filter with a window function that at tenuates the high frequencies [1]. This windowing is equivalent to radially smoothing the projection measurements with a spatially-invariant low-pass filter. In high resolution tomographs, such as positron emission tomography (PET) systems based on block detectors, large variations in detector efficiency lead to nonstat ionary variances. The equality of Poisson mean and variance also leads to an objectdependent variance nonuniformity. Spatially-invariant smoothing is subopt imal when the measurement statistics are nonstationary, since it treats all measurements equally. In this paper, we describe an information-weighted spline smoothing approach that accounts for the relative variances of different detectors. When applying spatially-invariant filters to projection da ta with nonuniform variance, one faces two unattract ive alternatives: oversmooth most of the measurements so as to minimize the noise due to the more variable data, or under-